Proof for D'alembert's solution

Question

I am totally clueless in how to go about the solution for part iii). I was able to answer the others. Normally I would find the derivatives then substitute. In regards to the initial conditions I'm lost. Someone please explain how I could go about the solution.

Answer 1

Are $F$ and $F_x$ continuous? By superposition, it suffices to show that $$v(x,t) = \frac{1}{2c} \int_{0}^{t} \int_{x-c(t-\tau)}^{x+c(t-\tau)} F(\xi, \tau) d\xi d\tau$$ satisfies the given equation and homogeneous versions of the initial condition. For more info see below:

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From Pinchover and Rubinstein:

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